Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Hochschild homology in a braided tensor category


Author: John C. Baez
Journal: Trans. Amer. Math. Soc. 344 (1994), 885-906
MSC: Primary 16W99; Secondary 16E40, 18G99
DOI: https://doi.org/10.1090/S0002-9947-1994-1240942-2
MathSciNet review: 1240942
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: An r-algebra is an algebra A over k equipped with a Yang-Baxter operator $ R:A \otimes A \to A \otimes A$ such that $ R(1 \otimes a) = a \otimes 1$, $ R(a \otimes 1) = 1 \otimes a$, and the quasitriangularity conditions $ R(m \otimes I) = (I \otimes m)(R \otimes I)(I \otimes R)$ and $ R(I \otimes m) = (m \otimes I)(I \otimes R)(R \otimes I)$ hold, where $ m:A \otimes A \to A$ is the multiplication map and $ I:A \to A$ is the identity. R-algebras arise naturally as algebra objects in a braided tensor category of k-modules (e.g., the category of representations of a quantum group). If $ m = m{R^2}$, then A is both a left and right module over the braided tensor product $ {A^e} = A\hat \otimes {A^{{\text{op}}}}$, where $ {A^{{\text{op}}}}$ is simply A equipped with the "opposite" multiplication map $ {m^{{\text{op}}}} = mR$. Moreover, there is an explicit chain complex computing the braided Hochschild homology $ {H^R}(A) = \operatorname{Tor}^{{A^e}}(A,A)$. When $ m = mR$ and $ {R^2} = {\text{id}}_{A \otimes A}$, this chain complex admits a generalized shuffle product, and there is a homomorphism from the r-commutative differential forms $ {\Omega _R}(A)$ to $ {H^R}(A)$.


References [Enhancements On Off] (What's this?)

  • [1] J. Baez, R-commutative geometry and quantization of Poisson algebras, Adv. Math. 95 (1992), 61-91. MR 1176153 (93j:58009)
  • [2] -, Differential calculi for quantum vector spaces with Hecke-type relations, Lett. Math. Phys. 23 (1991), 133-141. MR 1148505 (93c:58019)
  • [3] H. Cartan and S. Eilenberg, Homological algebra, Princeton Univ. Press, Princeton, N.J., 1956. MR 0077480 (17:1040e)
  • [4] A. Connes, Non-commutative differential geometry, Publ. Math. IHES 62 (1985), 257-360. MR 823176 (87i:58162)
  • [5] P. Deligne and J. Milne, Tannakian categories, Lecture Notes in Math., vol. 900, Springer, New York, 1982.
  • [6] V. Drinfeld, Quantum groups, Proc. Internat Congr. Math., 1986, pp. 798-820. MR 934283 (89f:17017)
  • [7] L. Faddeev, N. Reshetikhin, and L. Takhtajan, Quantization of Lie groups and Lie algebras, Leningrad Math. J. 1 (1990), 193-225. MR 1015339 (90j:17039)
  • [8] P. Feng and B. Tsygan, Hochschild and cyclic homology of quantum groups, Comm. Math. Phys. 140, 481-521. MR 1130695 (93e:19006)
  • [9] J. Fröhlich and F. Gabbiani, Braid group statistics in local quantum theory, preprint.
  • [10] P. Freyd and D. Yetter, Braided compact closed categories with applications to low dimensional topology, Adv. Math. 77 (1989), 156-182. MR 1020583 (91c:57019)
  • [11] G. Hochschild, B. Kostant and A. Rosenberg, Differential forms on regular affine algebras, Trans. Amer. Math. Soc. 102 (1962), 383-408. MR 0142598 (26:167)
  • [12] A. Joyal and R. Street, Braided monoidal categories, Mathematics Reports 86008, Macquarie University, 1986.
  • [13] -, The geometry of tensor calculus I, Adv. Math. 88 (1991), 55-112. MR 1113284 (92d:18011)
  • [14] M. Karoubi, Homologie cyclique et K théorie, Astérisque 149 (1987), 1-147. MR 913964 (89c:18019)
  • [15] L. Kauffman, On knots, Princeton Univ. Press, Princeton, N.J., 1986.
  • [16] -, Knots and physics, World Scientific, New Jersey, 1991.
  • [17] V. Lyubashenko, Hopf algebras and vector-symmetries, Uspehi Mat. Nauk 41 (1986), 185-186. MR 878344 (88c:58007)
  • [18] S. Mac Lane, Natural associativity and commutativity, Rice Univ. Studies 69 (1963), 28-46. MR 0170925 (30:1160)
  • [19] S. Majid, Quasitriangular Hopf algebras and Yang-Baxter equations, Internat J. Modern Phys. A 5 (1990), 1-91. MR 1027945 (90k:16008)
  • [20] Yu. Manin, Quantum groups and noncommutative geometry, Publ. du Centre de Récherches Math., Université de Montréal, Montréal, 1988. MR 1016381 (91e:17001)
  • [21] -, Topics in noncommutative geometry, Princeton Univ. Press, Princeton, N.J., 1991.
  • [22] W. Pusz, Twisted canonical anticommutation relations, Rep. Math. Phys. 27 (1989), 349-360. MR 1109004 (92i:81132)
  • [23] W. Pusz and S. Woronowicz, Twisted second quantization, Rep. Math. Phys. 27 (1989), 231-257. MR 1067498 (93c:81082)
  • [24] J.-L. Loday and D. Quillen, Cyclic homology and the Lie algebra homology of matrices, Comment. Math. Helv. 59 (1984), 565-591. MR 780077 (86i:17003)
  • [25] N. Reshetikhin and V. Turaev, Ribbon graphs and their invariants derived from quantum groups, Comm. Math. Phys. 127 (1990), 1-26. MR 1036112 (91c:57016)
  • [26] M. Rieffel, Noncommutative tori, a case study of noncommutative differential manifolds, Contemp. Math. 105 (1990), 191-211. MR 1047281 (91d:58012)
  • [27] N. Saavedra Rivano, Catégories Tannakiennes, Lecture Notes in Math., vol. 265, Springer, Berlin-Heidelberg, 1975.
  • [28] P. Seibt, Cyclic homology of algebras, World Scientific, Singapore, 1987. MR 938097 (89i:18013)
  • [29] B. Tsygan, Homology of matrix Lie algebras over rings and Hochschild homology, Uspehi Mat. Nauk 38 (1983), 217-218. MR 695483 (85i:17014)
  • [30] K.-H. Ulbrich, On Hopf algebras and rigid monoidal categories, Israel J. Math. 72 (1990), 252-256. MR 1098991 (92e:16031)
  • [31] J. Wess and B. Zumino, Covariant differential calculus on the quantum hyperplane, preprint. MR 1128150 (92g:58011)
  • [32] C. Yang and M. Ge, Braid group, knot theory, and statistical mechanics, World Scientific, 1989. MR 1062420 (92i:57011)
  • [33] B. Zumino, Deformation of the quantum mechanical phase space with bosonic or fermionic coordinates, preprint. MR 1110758 (93a:81087)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 16W99, 16E40, 18G99

Retrieve articles in all journals with MSC: 16W99, 16E40, 18G99


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1994-1240942-2
Article copyright: © Copyright 1994 American Mathematical Society

American Mathematical Society